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Plan

• 1.3

• 1.4

1.3

• Definition: A subset $W$ of a vector space $V$ over a field $F$ is called a subspace of $V$ if $W$ is a vector space over $F$ with the operations of addition and scalar multiplication defined on $V$.

• Problem: Normally, there are 8 properties you need to check. But it turns out you only need to check 4 of them. Which 4? Why?

• Problem: (Theorem 1.3) Let $V$ be a vector space and $W$ and subset of $V$. Then $W$ is a subspace of $V$ if and only if the following three conditions hold for the operations defined in $V$.

  1. $0\in W$.

  2. $x+y\in W$ whenever $x\in W$ and $y\in W$.

  3. $cx\in W$ whenever $c\in F$ and $x\in W$.

• Problem: Same problem as before, but replace conditions 2 and 3 with

  • $cx+y\in W$ whenever, $x,y\in W$ and $c\in F$.

• Problem: Give an example of a vector space $V$ and a subset $W$ of $V$ such that, $W$ is a vector space but $W$ is not a subspace of $V$.

• Problem: Show that the intersection of 2 subspaces is a subspace.

1.4

• Definition: Let $V$ be a vector space and $S$ a nonempty subset of $V$. A vector $v\in V$ is called a linear combination of vectors of $S$ if there exists a finite number of vectors $u_1,\ldots,u_n$ in $S$ and scalars $a_1,\ldots,a_n$ in $F$ such that $v=a_1u_1+\cdots+a_nu_n$.

• Problem: We denote the set of all linear combinations of $S$ by $\mathrm{span} S$. By convention, we define the span of the empty set to be the trivial subspace $\{0\}$. Prove that $\mathrm{span}(S)$ is always a subspace.

• Problem: Let $S\subseteq T$ be sets inside of a vector space $V$. Prove that $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(T)$.

• Problem: Prove that $\mathrm{span}(S)$ is a the smallest subspace containing $S$. (This gives an alternative definition of $\mathrm{span}(S)$ that turns out to be quite useful!)